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Theory of excitation energy transfer: from structure to function

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Abstract

This mini-review summarizes our current theoretical knowledge about excitation energy transfer in pigment–protein complexes. The challenge for theory lies in the complexity of these systems and in the fact that the pigment–pigment and the pigment–protein interactions are of equal magnitude. The first part of this review contains an introduction to the theory of light harvesting and to structure-based calculations of the parameters of the theory. The second part provides a discussion of the standard Förster and Redfield theories of excitation energy transfer, which are valid in the limit of weak and strong pigment–pigment coupling, respectively. Afterward, we provide a description of recent extensions of the standard theories and discuss challenging problems to be solved in the future.

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Notes

  1. In the language of molecular physics, an exciton is simply an excited state that residues on different molecules during its lifetime.

  2. Any polarizability that is slow compared to the electronic transition has no time to react. Therefore, the static part of the dielectric constant has no influence on the excitonic coupling.

  3. In a recent report (Renger et al. 2009) a possible range of 1.82–2.04 was estimated for the optical dielectric constant ε = n 2 of pigment–protein complexes.

  4. Interestingly, the inclusion of vibrational sidebands, resonance energy transfer narrowing and lifetime broadening into the theory used for the fit of optical spectra resulted in very similar site energies (Adolphs and Renger 2006) as the earlier fit of Louwe et al. (1997), where these effects were neglected.

  5. The \(\hbar \omega\) in Eq. 6 is the energy of the quanta of the vibrational mode with frequency ω, and J(ω) is the spectral density introduced earlier.

  6. The subscripts α and I were introduced to distinguish the lineshape of absorbance from that of fluorescence, respectively.

  7. \(|c_m^{(M)}|^2\) describes the probability to find pigment m excited in the Mth exciton state of the complex.

  8. As shown by Renger and Marcus (2003).

  9. The maximum of the distribution of energy gaps between the exciton states in the case of homodimers was 150 cm−1 and for heterodimers it was 400 cm−1 (Renger et al. 2007). At the latter energy the one-quantum dissipation function \(\tilde{C}^{(\rm Re)}(\omega)\) entering the Redfield rate constant in Eq. 10 is already small (see Fig. 8) and multi-vibrational quanta dissipation becomes dominant.

  10. The dynamic localization of exciton states by the pigment–protein coupling, which occurs when pigments are coupled weakly, even if they have the same site energy, is implicitly described by allowing delocalization only within the domains.

Abbreviations

BChl:

Bacteriochlorophyll

Chl:

Chlorophyll

CDC:

Charge density coupling

FMO:

Fenna–Matthews–Olson

TrEsp:

Transition charge from electrostatic potential

TDDFT:

Time-Dependent Density Functional Theory

XC:

Exchange correlation

WSCP:

Water soluble chlorophyll protein

PSI:

Photosystem I

PSII:

Photosystem II

LHCII:

Light harvesting complex of photosystem II

LH1:

Core light harvesting complex of purple bacteria

LH2:

Peripheral light harvesting complex of purple bacteria

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Acknowledgments

I would like to acknowledge support by the Deutsche Forschungsgemeinschaft through Emmy-Noether research grant RE 1610 and by the collaborative research centers Sfb 429 (TP A9) and Sfb 498 (TP A7).

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Authors and Affiliations

  1. Institut für Chemie und Biochemie, Freie Universität Berlin, Fabeckstrasse 36a, 14195, Berlin, Germany

    Thomas Renger

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Appendix

Appendix

Applications of TDC and TrEsp methods for coupling calculations

The TDC and TrEsp methods have been applied to calculate excitonic couplings, e.g., between the pigments in the light-harvesting complex II (LHC-II) of plants (Frahmke and Walla 2006), the light-harvesting complex 2 (LH2) of purple bacteria (Krueger et al 1998), the Fenna–Matthews–Olson (FMO) protein of green sulfur bacteria (Müh et al. 2007; Adolphs et al. 2008), photosystem II core complexes (Raszewski and Renger 2008), the special pairs in the reaction center of photosystem I of cyanobacteria, and the reaction center of purple bacteria (Madjet et al. 2006).

Structure-based calculations of site energies

Early quantum chemical calculations of site energies used a cutoff radius to include every amino acid with an atom within this radius of any pigment atom in the quantum chemical calculation of the optical transition energy of this pigment. This method was applied to the FMO-protein (Gudowska-Nowak et al. 1990) and photosystem I (Damjanovic et al. 2002a). The drawback of this method is the neglect of long-range electrostatic interactions of the pigments with protein parts outside the cutoff radius. In an alternative quantum chemical method the whole photosystem I protein was included by atomic partial charges into the quantum chemical calculation of pigment transition energies (Yin et al. 2007). The drawback of this method is the artificial distortion of the electronic wavefunction of the pigments by the external charges (the so-called electron leakage problem), which has its origin in the neglect of Pauli repulsion (Laio et al. 2002). These problems could be circumvented by a combined quantum chemical/electrostatic approach (Müh et al 2007; Adolphs et al. 2008), which was successfully applied to the FMO protein, as described in the main text.

Lineshape functions for absorbance and fluorescence of a two-level system coupled to a continuum of harmonic oscillators

The line shape functions are obtained as (Lax 1952; May and Kühn (2000); Renger and Holzwarth 2008)

$$ D_{\alpha/I}^{(k)}(\omega)={\hbox{e}}^{-G(0)} \left(\delta(\omega-\omega_{eg}^{(k)}) + \phi_k(\pm (\omega-\omega_{eg}^{(k)})) \right) $$
(15)

where the first part on the r.h.s. describes the purely electronic excitation (the 0-0 transition) at frequency \(\omega_{eg}^{(k)}\) and the second part the vibrational sideband

$$ \phi_k(\pm (\omega-\omega_{eg}^{(k)}))= \frac{1} {2\pi}\int\limits_{-\infty}^{\infty}\hbox{d}t\,\hbox{e}^{\pm {i} (\omega-\omega_{eg}^{(k)}) t} (\hbox{e}^{G(t)}-1). $$
(16)

The time-dependent function G(t) contains the spectral density of the pigment–protein coupling J(ω) and the mean number of vibrational quanta that are excited at a given temperature (Bose-Einstein distribution function) n(ω),

$$ G(t)=\int\limits_{0}^{\infty}\hbox{d}\omega \{ (1+n(\omega)) J(\omega)\,\hbox{e}^{-{i}\omega t}+ n(\omega)J(\omega)\hbox{e}^{{\rm i}\omega t}\}, $$
(17)
$$ n(\omega)=\frac{1}{\hbox{e}^{\hbar \omega/kT}-1}. $$
(18)

Applications of multi-level Redfield theory

Multi-level Redfield theory was used to describe excitation energy transfer, e.g., between the strongly coupled BChls in the LH2 antenna complex (Kühn and Sundström 1997; Kühn et al. 2002), in the reaction center of photosystem II (Leegwater et al 1997; Prokhorenko and Holzwarth 2000; Renger and Marcus 2002b), in the FMO-protein (Renger and May 1998; Adolphs and Renger 2006; Müh et al. 2007), in photosystem I core complexes (Brüggemann et al. 2004), in the LHC-II of green plants (Novoderezhkin et al. 2003), in the core light-harvesting complex LH1 of purple bacteria (Novoderezhkin and van Grondelle 2002), in the WSCP complex (Renger et al. 2007), in the domains of strongly coupled Chls of photosystem II core complexes (Raszewski and Renger 2008), and in model dimers (Kjellberg and Pullerits 2006; Yang and Fleming 2002).

Modified Redfield theory

The rate constant in modified Redfield theory is given as (Renger and Marcus 2003)

$$ k_{M\rightarrow N}=\int\limits_{-\infty}^{\infty}{\hbox{d}}\tau {\hbox{e}}^{{\rm i}\omega_{MN}\tau}\,\hbox{e}^{\phi_{MN}(\tau)-\phi_{MN}(0)} \times\left[\left(\frac{\lambda_{MN}}{\hbar}+G_{MN}(\tau)\right)^2+F_{MN}(\tau)\right], $$
(19)

where the time-dependent functions

$$ \phi_{MN}(t)= a_{MN} \phi_0(t), $$
(20)
$$ G_{MN}(t)=b_{MN} \phi_1(t), $$
(21)
$$ F_{MN}(t)=\gamma_{MN} \phi_2(t) $$
(22)

are related to the spectral density J(ω) via the function ϕ n (t),

$$ \phi_n(t)=\int\limits_{-\infty}^{\infty} \hbox{d}\omega\,\hbox{e}^{-{i}\omega t}(1+n(\omega))\omega^n (J(\omega)-J(-\omega)). $$
(23)

The time-independent part in the integrand in Eq. 19 is

$$ \lambda_{MN}=d_{MN} \lambda, $$
(24)

with the reorganization energy λ in Eq. 6. The coefficient γ MN was introduced in Eq. 12.

The coefficients a MN , b MN , d MN of modified Redfield theory were originally derived in the limit \(R_c \rightarrow \infty\) (Renger and Marcus 2003). The more general case of finite R c was considered later and yields the coefficients (Raszewski et al. 2005)

$$ a_{MN}=\sum_{ij} ((c_i^{(M)})^2(c_j^{(M)})^2+(c_i^{(N)})^2(c_j^{(N)})^2 -2(c_i^{(M)})^2(c_j^{(N)})^2) e^{-R_{ij}/R_c} $$
(25)
$$ b_{MN}=\sum_{ij} ((c_i^{(M)})^2 - (c_i^{(N)})^2) c_j^{(M)} c_j^{(N)} e^{-R_{ij}/R_c} $$
(26)
$$ d_{MN}=\sum_{ij} ((c_i^{(M)})^2 + (c_i^{(N)})^2) c_j^{(M)} c_j^{(N)} e^{-R_{ij}/R_c}. $$
(27)

Lineshape functions entering Generalized Förster theory

The multi-level lineshape functions, obtained with non-Markovian partial ordering prescription theory for absorbance and fluorescence read (Renger and Marcus 2002a; Raszewski and Renger 2008)

$$ D_{M_d}(\omega)=\frac{1} {2\pi} \int\limits_{-\infty}^{\infty} \hbox{d}t\,\hbox{e}^{{i}(\omega-\tilde{\omega}_{M_d})t} \hbox{e}^{G_{M_d}(t)-G_{M_d}(0)} \hbox{e}^{-|t|/\tau_{M_d}}, $$
(28)

and

$$ D_{M_d}^\prime (\omega)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty} \hbox{d}t \hbox{e}^{-{\rm i}(\omega-\tilde{\omega}_{M_d})t} \hbox{e}^{G_{M_d}(t)-G_{M_d}(0)} \hbox{e}^{-|t|/\tau_{M_d}}, $$
(29)

respectively, that contain the time-dependent function

$$ G_{M_d}(t)=\gamma_{M_d M_d} G(t), $$
(30)

where the \(\gamma_{M_d M_d}\) and G(t) are given in Eqs. 12 and 17, respectively, and the frequency \(\tilde{\omega}_{M_d}\) takes into account a shift of the exciton transition frequency \(\omega_{M_d}={\mathcal E}_{M_d}/\hbar\) by the pigment–protein coupling (Renger and May 2000; Renger and Marcus 2002a),

$$ \tilde{\omega}_{M_d}=\omega_{M_d}-\gamma_{M_dM_d}\lambda/\hbar +\sum_{K_d\ne M_d} \gamma_{M_dK_d}\times \wp \int\limits_{-\infty}^{\infty}\hbox{d}{\omega} \frac{\omega^2\left\{(1+n(\omega))J(\omega)+ n(-\omega)J(-\omega) \right\}}{\omega_{M_dK_d}-{\omega}}, $$
(31)

where λ is given in Eq. 6, \(\gamma_{M_dM_d}\) in Eq. 12, and ℘ denotes the principal part of the integral. As noted before, it holds that J(ω) = 0 for ω < 0. The \(\tau_{M_d}=\frac{1}{2}\sum_{N_d} k_{M_d\rightarrow N_d}\) describes life time broadening due to exciton relaxation with the Redfield rate constants \(k_{M_d\rightarrow N_d}\) given in Eq. 10.

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Renger, T. Theory of excitation energy transfer: from structure to function. Photosynth Res 102, 471–485 (2009). https://doi.org/10.1007/s11120-009-9472-9

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